My card is the only thing holding me back from moving to a little newer range of games; can anyone recommend one that might work with my case / power supply? Or tell me if it is even possible to add a new graphics card to this set up?

If you plan on gaming, anything under a (nvidia) 9600gt or (ati) 4670 is a waste. Those are the bare minimum I'd suggest for a decent gaming experience. It's possible a 4670 would work with your PSU, but I'd probably still upgrade it.
What's your budget for the upgrade? We could probably find ya a good deal somewhere on a combo.

they'd need to know what the +12V Amp ratings were on your Power supply, before they can say for sure that it'd work, but knowing how most prebuilt PC's are and given its a 300W unit, I'd say it might not handle it, or if It could I wouldn't chance it. if the PSU can't handle the load it could die and take most of your system with it.

$40 can get you a decent quality brand PSU that can handle the load, and another $60-70 for a decent budget gaming card.

Looking at the case, it looks, note I said looks, like it should take a standard ATX design PSU, but with out some measurements or at least some images of the PSU and back of case, couldn't say for sure.

most likely though it should have a standard ATX design or at least Micro ATX Power Supply.

Listed from Least to Best in performance, the first should be bare minimum for any sort of gaming card, with the exception of maybe a Geforce GT 220, which should perform similarly to the version of the 9600GSO listed here (this one is a version that only has 48 Stream Processor cores, the regular GSO has 96)

I should note these are on the lower end of the GPU's out today. They will handle most games on moderate settings. But if your trying to max out crysis or something like that, then these probably aren't gonna cut it for you.
Good luck, hope I helped!